Editing to a Graph of Given Degrees

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8894)

Abstract

We consider the Editing to a Graph of Given Degrees problem that for a graph \(G\), non-negative integers \(d,k\) and a function \(\delta :V(G)\rightarrow \{1,\ldots ,d\}\), asks whether it is possible to obtain a graph \(G'\) from \(G\) such that the degree of \(v\) is \(\delta (v)\) for any vertex \(v\) by at most \(k\) vertex or edge deletions or edge additions. We construct an FPT-algorithm for Editing to a Graph of Given Degrees parameterized by \(d+k\). We complement this result by showing that the problem has no polynomial kernel unless \(\mathrm{{NP}}\subseteq \mathrm{{coNP}}/\mathrm{{poly}}\).

References

  1. 1.
    Alon, N., Shapira, A., Sudakov, B.: Additive approximation for edge-deletion problems. In: FOCS, pp. 419–428. IEEE Computer Society (2005)Google Scholar
  2. 2.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Cross-composition: a new technique for kernelization lower bounds. In: STACS. LIPIcs, vol. 9, pp. 165–176. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2011)Google Scholar
  4. 4.
    Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Kernelization lower bounds by cross-composition. CoRR abs/1206.5941 (2012)
  5. 5.
    Burzyn, P., Bonomo, F., Durán, G.: NP-completeness results for edge modification problems. Discrete Appl. Math. 154(13), 1824–1844 (2006)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58(4), 171–176 (1996)CrossRefMATHGoogle Scholar
  7. 7.
    Cai, L., Chan, S.M., Chan, S.O.: Random separation: a new method for solving fixed-cardinality optimization problems. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 239–250. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Cai, L., Yang, B.: Parameterized complexity of even/odd subgraph problems. J. Discrete Algorithms 9(3), 231–240 (2011)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Cygan, M., Marx, D., Pilipczuk, M., Pilipczuk, M., Schlotter, I.: Parameterized complexity of Eulerian deletion problems. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 131–142. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2006)Google Scholar
  11. 11.
    Froese, V., Nichterlein, A., Niedermeier, R.: Win-win kernelization for degree sequence completion problems. CoRR abs/1404.5432 (2014)
  12. 12.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)MATHGoogle Scholar
  13. 13.
    Golovach, P.A.: Editing to a connected graph of given degrees. CoRR abs/1308.1802 (2013)
  14. 14.
    Golovach, P.A.: Editing to a graph of given degrees. CoRR abs/1311.4768 (2013)
  15. 15.
    Khot, S., Raman, V.: Parameterized complexity of finding subgraphs with hereditary properties. Theor. Comput. Sci. 289(2), 997–1008 (2002)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is np-complete. J. Comput. Syst. Sci. 20(2), 219–230 (1980)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Mathieson, L., Szeider, S.: Editing graphs to satisfy degree constraints: A parameterized approach. J. Comput. Syst. Sci. 78(1), 179–191 (2012)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Moser, H., Thilikos, D.M.: Parameterized complexity of finding regular induced subgraphs. J. Discrete Algorithms 7(2), 181–190 (2009)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Naor, M., Schulman, L., Srinivasan, A.: Splitters and near-optimal derandomization. In: 36th Annual Symposium on Foundations of Computer Science (FOCS 1995), pp. 182–191. IEEE (1995)Google Scholar
  20. 20.
    Natanzon, A., Shamir, R., Sharan, R.: Complexity classification of some edge modification problems. Discrete Appl. Math. 113(1), 109–128 (2001)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms, Oxford Lecture Series in Mathematics and Its Applications, vol. 31. Oxford University Press, Oxford (2006)Google Scholar
  22. 22.
    Seese, D.: Linear time computable problems and first-order descriptions. Math. Struct. Comput. Sci. 6(6), 505–526 (1996)MATHMathSciNetGoogle Scholar
  23. 23.
    Yannakakis, M.: Node- and edge-deletion NP-complete problems. In: Lipton, R.J., Burkhard, W.A., Savitch, W.J., Friedman, E.P., Aho, A.V. (eds.) STOC, pp. 253–264. ACM (1978)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.Steklov Institute of Mathematics at St. PetersburgRussian Academy of SciencesSt. PetersburgRussia

Personalised recommendations